| Property | Level | Notes |
|---|---|---|
| a_b | Species | Intercept of intrinsic growth rate - temperature function of species i. |
| b_{opt,i} | Species | Temperature at which intrinsic growth rate is highest for species i. |
| s_i | Species | Width of intrinsic growth rate - temperature function of species i. |
| a_d | Species | Intercept of death rate - temperature function of species i. |
| z_i | Species | Slope of death rate - temperature function of species i. |
| alpha_{i,j} | Interaction | Strength of interspecific effect of species j on species i. |
Species interactions determine the importance of response diversity for community stability to pulse disturbances
1 Introduction
In the context of a pulse disturbance we ask:
- will fundamental niche-based responses, such as those measured according to Ross et al (2023), explain variation in community stability?
- are realised responses able to explain more variation in community stability?
We explore these questions using a simulation model of a multi-species model, in which species intrinsic growth rate is a bell-shaped function of temperature, and each species can have a different temperature optimum. A pulse perturbation is created by briefly setting temperature lower than the control temperature, and then returning it to control temperature.
We then manipulate the strength of inter-specific interactions and examine how that affects the explanatory power of the two types of response trait.
2 Methods
2.1 The multispecies model
We have \(S\) species that can be interacting and whose vital rates are temperature dependent. We assume density-dependent birth rate, (\(B\)), and death rate (\(D\)) in a discrete-time version of the classical Lotka–Volterra model [@de2013predicting; @vasseur2020impact] to get instantaneous growth rate, \(\tilde{r}_{i}(t)\), for species, \(i\), in year \(t\):
\[\begin{equation}\label{eq.r} \tilde{r}_{i}(t) = ln N_{i}(t+1) - ln N_{i}(t) = B(N_{i}(t),N_{j}(t),T(t)) - D(N_{i}(t),N_{j}(t),T(t))(\#eq:r) \end{equation}\]Here, \(N(t)\) represents the biomass at year \(t\), and \(i,j\) are indices for two different species.
The per-capita birth and death rates for \(i^{th}\) species are represented as:
\[\begin{equation}\label{eq.B} B_{i} = b_{0,i}(T)-\beta (N_{i}+\sum_{i \neq j = 1}^{S} \alpha_{ij}N_{j})(\#eq:B) \end{equation}\] \[\begin{equation}\label{eq.D} D_{i} = d_{0,i}(T)-\delta (N_{i}+\sum_{i \neq j = 1}^{S} \alpha_{ij}N_{j})(\#eq:D) \end{equation}\]where, \(\beta\) and \(\delta\) are density-dependent constants, \(\alpha_{i,j}\) is the competition coefficient between species \(i\) and \(j\), and
\[\begin{equation}\label{eq.b0} b_{0,i}(T) = a_{b} e^{-(T-b_{opt,i})^2/s_{i}} (\#eq:b0) \end{equation}\] \[\begin{equation}\label{eq.d0} d_{0,i}(T) = a_{d} e^{z_{i}T} (\#eq:d0) \end{equation}\]with \(a_{b}\), \(a_{d}\) as intercepts, and for \(i^{th}\) species, \(b_{opt,i}\) is the temperature that optimizes birth rate, \(s_{i}\) governs the breadth of the birth function, \(z_{i}\) scales the effect of temperature (in °C) to mimic the Arrhenius relationship.
Substituting Eqs. (\(\ref{eq.B}\)) - (\(\ref{eq.d0}\)) in Eq. (\(\ref{eq.r}\)), we get the following,
\[\begin{equation}\label{eq.r2} \tilde{r}_{i}(t) = r_{m,i} \left( 1-\dfrac{\sum_{i, j = 1}^{S} \alpha_{ij}N_{j}}{K_{i}} \right) (\#eq:r2) \end{equation}\]where, \(\alpha_{ii} = 1\), \(r_{m,i} = (b_{0,i} - d_{0,i})\) is the intrinsic (maximum) rate of increase and \(K_{i} = r_{m,i}/(\beta+\delta)\) is the carrying capacity for the \(i^{th}\) species, respectively.
If abundance drops very low then it is reset to 1. May need to change this, or at least present a good ecological justification.
2.2 Model parameters
The following need to be specified for a community (in addition to the number of species):
2.3 Experimental treatments
2.3.1 Pulse disturbance treatment
The context of our entire study is a pulse perturbation. Temperature is constant in one treatment, and is pulse perturbed in another. For simplicity, we always make the perturbed temperature lower than the control temperature. We do not manipulate or change anything about the pulse perturbation. The graph below shows the control temperature in black and the pulse in red. Simulations are run for 10’000 time steps before the pulse is applied and run for another 1’000 after the pulse starts.
2.3.2 Creating a community
In order to create a community we define a set of \(S\) species and in the following only two things differ among the species:
- The optimum temperature for growth, such that some species have maximum intrinsic growth rate at higher temperatures and some have maximum intrinsic growth at lower temperatures. The optimum temperature for growth for each species in a community is drawn from a uniform distribution with a specific mean and range.
- The interspecific interactions in the interaction matrix \(alpha_{i,j}\). These are drawn from a normal distribution with mean zero and standard deviation \(sd(alpha_{i,j})\).
2.3.3 Community composition treatment
We create communities that vary in their composition by changing the mean of the uniform distribution from which species’ optimum temperature for growth are drawn. If we set the mean low, close to the temperature of the perturbation, then the perturbation as a positive effect on many of the species intrinsic growth rates. In contrast, if we set the mean close to the control temperature then the perturbation has mostly negative effects on the species’ intrinsic growth rates. We show some communities in the Results section of this report.
2.3.4 Strength of species interactions treatment
When we set \(sd(alpha_{i,j})\) = 0 then there are no interspecific interactions. Larger values of \(sd(alpha_{i,j})\) mean that interspecific interactions are stronger.
2.4 Analyses
2.4.1 Species responses
We calculated two types of species responses. First was the species fundamental response based on the temperature response curve of a species (i.e., as in Ross et al (2023)). For this, we calculated the effect of the perturbation on intrinsic growth rate (perturbation temperature intrinsic growth rate - control temperature intrinsic growth rate). This is analogous to the calculating the first derivative (slope) of the temperature response curve.
Second was a realised response based on the observed response of the species to the pulse perturbation in the community context. Hence this response depends on the direct effect of the pulse on a species, and also the indirect effects via other species in the community.
2.4.2 Community response diversity
Response diversity can be measured in several ways, including with the two measures response dissimilarity and response divergence (Ross et al., 2023), alongside the mean species response. Response diversity was assessed using two complementary components: the dissimilarity metric and the divergence metric (Ross et al., 2023). Both measures of response diversity were assessed from fundamental species responses (IGR effect), i.e. fundamental response diversity and from species realised responses (AUC.RR), i.e. realised response diversity. Currently we calculate:
2.4.3 Community stability
This was calculated as the effect of the pulse perturbation on the total community biomass. This effect varied through time (e.g., often decreased monotonically after the pulse ended). The effect was calculated as the control - perturbed abundance and could conceivably be sometimes positive and sometimes negative. To measure community stability, we used the Overall Ecological Vulnerability index (OEV, Urrutia-Cordero et al., 2021).
3 Results
3.1 Species responses and community stability
In the following various graphs is the relationship between community stability (y-axis) and the mean of species responses (realised or fundamental) (x-axis) .
Each panel is for a different strength of interspecific interaction.
Examples 4 has relatively weak interspecific interaction, and example 5 has relatively strong interspecific interaction. The black boxes in some of the individual species responses are the addition of 1 abundance unit when the abundance drop too low (see last sentence in section Section 2.1).
3.1.1 Example 1
No inter-specific interactions (sd(alpha_ij) = 0) and low average optimum temperature (mean(b_opt) = 17), so the pulse has positive effect on intrinsic growth rate of most species.
| species_id | species_RR_AUC | igr_pert_effect |
|---|---|---|
| Spp1 | 121.04475 | 0.2825863 |
| Spp2 | -13.34736 | -0.0509040 |
| Spp3 | 46.77071 | 0.1499659 |
| Spp4 | 20.68775 | 0.0689394 |
| Spp5 | -17.62187 | -0.0697486 |
| Spp6 | 146.44123 | 0.2977998 |
| Spp7 | -22.43094 | -0.0935146 |
| Spp8 | 34.62691 | 0.1132464 |
| Spp9 | 123.16240 | 0.2847423 |
| Spp10 | 139.45864 | 0.2963434 |
3.1.2 Example 2
No inter-specific interactions (sd(alpha_ij) = 0) and high average optimum temperature (mean(b_opt) = 20), so the pulse has negative effect on intrinsic growth rate of most species.
| species_id | species_RR_AUC | igr_pert_effect |
|---|---|---|
| Spp1 | -1.522639 | -0.0054039 |
| Spp2 | 8.069996 | 0.0276408 |
| Spp3 | -35.644817 | -0.1894834 |
| Spp4 | 30.304615 | 0.0996591 |
| Spp5 | -34.804671 | -0.1808283 |
| Spp6 | 14.681084 | 0.0494640 |
| Spp7 | -42.720835 | -0.2977818 |
| Spp8 | -38.019597 | -0.2178036 |
| Spp9 | -42.660936 | -0.2977212 |
| Spp10 | -27.038076 | -0.1200976 |
3.1.3 Example 3
No inter-specific interactions (sd(alpha_ij) = 0) and intermediate average optimum temperature (mean(b_opt) = 19), so the pulse has negative effect on intrinsic growth rate of some species, and positive effect on others.
| species_id | species_RR_AUC | igr_pert_effect |
|---|---|---|
| Spp1 | -25.02691 | -0.1079195 |
| Spp2 | 58.63932 | 0.1822471 |
| Spp3 | 34.71376 | 0.1135175 |
| Spp4 | -40.99210 | -0.2655814 |
| Spp5 | 35.96048 | 0.1173985 |
| Spp6 | 58.94636 | 0.1830210 |
| Spp7 | -14.65281 | -0.0564766 |
| Spp8 | -39.19559 | -0.2346461 |
| Spp9 | -40.59770 | -0.2581594 |
| Spp10 | -38.92272 | -0.2305311 |
3.1.4 Example 4
“Weak” inter-specific interactions (sd(alpha_ij) = 0.1) and intermediate average optimum temperature (mean(b_opt) = 19), so the pulse has negative effect on intrinsic growth rate of some species, and positive effect on others.
| species_id | species_RR_AUC | igr_pert_effect |
|---|---|---|
| Spp1 | -39.198088 | -0.2208045 |
| Spp2 | 159.022424 | -0.0142174 |
| Spp3 | -29.236368 | -0.1497255 |
| Spp4 | NA | 0.0721339 |
| Spp5 | 110.043055 | -0.0259111 |
| Spp6 | -43.099162 | -0.2428524 |
| Spp7 | NA | 0.1523334 |
| Spp8 | 2.603148 | -0.0230214 |
| Spp9 | -43.938091 | -0.2810086 |
| Spp10 | -44.680622 | -0.2750190 |
3.1.5 Example 5
“Strong* inter-specific interactions (sd(alpha_ij) = 0.3) and intermediate average optimum temperature (mean(b_opt) = 19), so the pulse has negative effect on intrinsic growth rate of some species, and positive effect on others.
| species_id | species_RR_AUC | igr_pert_effect |
|---|---|---|
| Spp1 | NA | 0.1198007 |
| Spp2 | -38.57910 | -0.2121045 |
| Spp3 | NA | 0.1766950 |
| Spp4 | NA | 0.0720570 |
| Spp5 | -35.01856 | -0.1907665 |
| Spp6 | NA | -0.0574842 |
| Spp7 | -37.65322 | -0.2107163 |
| Spp8 | 114.58998 | -0.0754677 |
| Spp9 | -36.27415 | -0.1968421 |
| Spp10 | -41.03496 | -0.2264514 |
3.1.6 Code for checking a specific case
Comm-1241-rep-1
| species_id | species_RR_AUC | igr_pert_effect |
|---|---|---|
| Spp1 | -29.549626 | -0.1798428 |
| Spp2 | -49.120164 | -0.2808470 |
| Spp3 | 52.360835 | -0.0244940 |
| Spp4 | -44.409931 | -0.2776526 |
| Spp5 | NA | -0.0341239 |
| Spp6 | NA | -0.0409027 |
| Spp7 | NA | -0.1232092 |
| Spp8 | -28.636600 | -0.1379287 |
| Spp9 | -9.084879 | -0.0894503 |
| Spp10 | -45.893623 | -0.2437334 |
4 Appendix
4.1 Check various relationships
